Biology, asked by kumarji5519, 1 year ago

Cumulative distribution calculation in psd analysis

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Answered by yenula

Answer:

The most widely used method of describing particle size distributions are D values. The D10, D50 and D90 are commonly used to represent the midpoint and range of the particle sizes of a given sample. Particle size distributions have been traditionally calculated based on sieve analysis results, creating an S-curve of cumulative mass retained against sieve mesh size, and calculating the intercepts for 10%, 50% and 90% mass. A new approach is required to calculate particle size distributions from laser and imaging technologies however, as these methods do not measure the mass of particles.

Calculating D-values

D Value chart

Computing D-values for a particle size distribution relies on modelling all particles as spheres. This is convenient when the majority of particles in a sample are relatively close to spherical in shape, but can cause problems when equivalent diameters must be used for more arbitrarily shaped particles. Several different attributes can be chosen to determine the size of an "equivalent sphere". Most commonly particles are modelled as spheres of equivalent volume, which assuming constant density, may be considered interchangeable with mass.

A D-value can be thought of as a “mass division diameter”. It is the diameter which, when all particles in a sample are arranged in order of ascending mass, divides the sample's mass into specified percentages. The percentage mass below the diameter of interest is the number expressed after the "D".

For example the D10 diameter is the diameter at which 10% of a sample's mass is comprised of smaller particles, and the D50 is the diameter at which 50% of a sample's mass is comprised of smaller particles. The D50 is also known as the "mass median diameter" as it divides the sample equally by mass.

This can be most clearly demonstrated using the particle size distribution graph below. This shows the particle volume distribution and cumulative volume of the sample. The volume distribution shows the particles in a given size range by percentage of the total sample volume, while the cumulative volume curve tracks the total volume of all size ranges as they approach 100%.

As the graph shows, the D10, D50 and D90 are given by the X axis (diameter) value where the cumulative volume curve crosses 10%, 50% and 90% on the Y axis.

Computing D Values Graph 2

Computing D-values from Optical Particle Size Measurement Systems

While D-values are based on a division of the mass of a sample by diameter, the actual mass of the particles or the sample does not need to be known. A relative mass is sufficient as D-values are concerned only with a ratio of masses. This allows the optical measurement systems discussed above to be used without any need for sample weighing.

From the diameter values obtained for each particle a relative mass can be assigned.

Mass of a sphere= π/6 d3 ρ

Assuming that ρ is constant for all particles and cancelling all constants from the equation:

Relative mass= d3

Each particle's diameter is therefore cubed to give its relative mass. These values can be summed to calculate the total relative mass of the sample measured. The values may then be arranged in ascending order and added iteratively until the total reaches 10%, 50% or 90% of the total relative mass of the sample. The corresponding D value for each of these is the diameter of the last particle added to reach the required mass percentage.

For a guide to particle sizing techniques and definitions used within the pharmaceutical industry download the PDF below.

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Tuomas Ervasti

School of Pharmacy, University of Eastern Finland